Compute and interpret 95% confidence intervals

Answer all parts. Show formulas and intermediate values. A) An A/B test measures conversion: Variant A has 410 conversions out of 5000 visitors; Variant B has 470 conversions out of 5100 visitors. Compute a two-sided 95% confidence interval for (p_B − p_A) using (i) the normal approximation with unpooled standard error and (ii) the Wilson score (Newcombe) method. State whether B is statistically different from A at α=0.05 under each method. Then, if you were actually testing 3 variants (B, C, D) against A and wanted family-wise α=0.05 via Bonferroni, what per-comparison confidence level would you use, and would the normal-approximation conclusion for B change? B) For mean session time (minutes), a sample of n=40 yields mean 8.2 and sample standard deviation 2.4. Compute the 95% confidence interval for the population mean using the t-distribution; report the margin of error using t_{0.975,39}. C) What minimum per-group sample size is required to estimate a single proportion with 95% confidence such that the half-width of the interval is at most 0.01, using the conservative p=0.5 assumption? Provide the formula and result.